Adaptive tracking control of sprott h system
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This paper presents an investigation into the tracking control of the Sprott-H chaotic system using new nonlinear control laws that account for unknown parameters. The adaptive generalized backstepping method is applied to achieve effective control over the system, demonstrated through numerical simulations. Results show the proposed method successfully addresses the tracking of desired input signals.
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013
ADAPTIVE TRACKING CONTROL OF SPROTT-H
SYSTEM
Masoud Taleb Ziabari 1 and Ali Reza Sahab 2
1
2
Faculty of Engineering, Computer Engineering Group, Ahrar University, Rasht, Iran.
Faculty of Engineering, Electrical Engineering Group, Islamic Azad University, Lahijan
Branch, Iran.
Abstract
This paper investigates the tracking problem of Sprott-H chaotic system which is three-dimensional chaotic
systems discovered by Sprott (1994). New nonlinear control laws are derived for the tracking problem of
uncertain Sprott-H chaotic system with unknown parameters. The adaptive generalized backstepping
method is applied to control of uncertain Sprott-H chaotic system. Numerical simulations are presented to
demonstrate the effectiveness of the control schemes.
KEYWORDS
Sprott-H Chaotic System, Uncertain, Adaptive Generalized Backstepping Method, Control.
1. INTRODUCTION
In recent years, various controllers have been proposed to achieve the control of chaotic systems
[3-9]. The adaptive synchronization of an uncertain modified hyperchaotic Lü system was
investigated in [10]. In [11], the output regulation problem for the Sprott-G chaotic system (1994)
has been studied in detail. The stabilization and synchronization of the hyperchaotic Cai system
with unknown system parameters was applied by adaptive control theory. The feedback
controllers for control of the simplified Lorenz system was investigated in [13]. [14] derive state
feedback controllers for the output regulation problem of the Sprott-H chaotic system (1994). In
[15], active controller has been designed to solve the output regulation problem for the Sprott-K
chaotic system (1994) and a complete solution for the tracking of constant reference signals (setpoint signals). [16] applied adaptive control theory for the control and synchronization of the
Sprott H chaotic system (Sprott, 1994) with unknown system parameters.
The rest of the paper is organized as follows: In section 2, Sprott-H system is presented. In
section 3, the generalized backstepping method is studied. In section 4, the generalized
backstepping controller is designed for racking any desired inputs. In section 5, Represents
simulation results. Finally, in section 6, Provides conclusion of this work.
DOI : 10.5121/ijitmc.2013.1409
83](https://image.slidesharecdn.com/adaptivetrackingcontrolofsprott-hsystem-131207003851-phpapp02/85/Adaptive-tracking-control-of-sprott-h-system-1-320.jpg){kind=tracking}
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013
2. SYSTEM DESCRIPTION
The three-dimensional Sprott-H chaotic system [17] described by the dynamics
̇
̇
(1)
̇
Where
are positive constants and
are variables of the system, when
system (1) is chaotic. See Figure 1 and Figure 2.
, the
6
x
y
z
Trajectory of States
4
2
0
-2
-4
-6
0
100
200
300
400
500
600
Time (sec)
700
800
900
1000
Figure 1. Time response of the system (1).
2
1
z
0
-1
-2
-3
6
4
2
0
2
-2
0
y
-4
-2
-6
x
Figure 2. Phase portraits of the hyperchaotic attractors (1).
3. GENERALIZED BACKSTEPPING METHOD
Generalized backstepping method [7-9] is applied to nonlinear systems as follow
{
̇
̇
Where
( )
( )
(
)
(
and
[
(2)
)
]
. Suppose the function ( ) is the lyapunov function.
84](https://image.slidesharecdn.com/adaptivetrackingcontrolofsprott-hsystem-131207003851-phpapp02/85/Adaptive-tracking-control-of-sprott-h-system-2-320.jpg){kind=tracking}
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013
( )
∑
(3)
The control signal and the extended lyapunov function of system (2) are obtained by equations
(4),(5).
(
(
∑
{
)
)
∑
[ ( )
∑
( )
∑
∑
[
∑
[
( ) ]
( )]
(
}
)
( )]
(4)
(5)
4. TRACKING OF CHAOTIC SYSTEM
First, we will add control laws
and the desired trajectory ( ). ̅
( ̅
̇
to the equation (1) let ̅ be the deviation between the output
( )
)
̇
(6)
̇̅
̅
̇
Stabilization of the state: In order to use the theorem, it is sufficient to establish equation (7).
(
̅)
̇
(
̅)
(7)
According to the theorem, the control signals will be obtained from the equations (8).
(
)
(
) ̂
̂
(
)
(8)
The parameters
are unknown and ̂ ̂ are respectively estimated values of parameters
which are updated by following equation.
̂̇
̂̇
̂
(9)
̂
And Lyapunov function as
(
)
(
̅
)
(
)
(10)
we select the gains of controllers (8) in the following form
(11)
5. NUMERICAL SIMULATION
This section presents numerical simulations Sprott-H chaotic system. The generalized
backstepping method (GBM) is used as an approach to control chaos in Sprott-H chaotic system.
( )
( )
The initial values of the Sprott-H chaotic system are ( )
.
Case 1 : Tracking ( )
,
85](https://image.slidesharecdn.com/adaptivetrackingcontrolofsprott-hsystem-131207003851-phpapp02/85/Adaptive-tracking-control-of-sprott-h-system-3-320.jpg){kind=tracking}
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013
6. CONCLUSIONS
In this paper, we applied adaptive generalized backstepping control for the control of the Sprott-H
chaotic system with unknown system parameters. The tracking problem considered for the SprottH chaotic system was for the tracking of any desired input signals. Numerical simulations show
that the proposed method work effectively.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
Chao-Chung Peng, Chieh-Li Chen. Robust chaotic control of Lorenz system by backstepping design.
Chaos, Solitons and Fractals 37 (2008) 598–608.
Cheng-Chi Wang, Neng-Sheng Pai, Her-Terng Yau. Chaos control in AFM system using sliding
mode control by backstepping design. Commun Nonlinear Sci Numer Simulat 15 (2010) 741–751.
Faqiang Wang, Chongxin Liu. A new criterion for chaos and hyperchaos synchronization using linear
feedback control. Physics Letters A 360 (2006) 274–278.
Yongguang Yu, Suochun Zhang. Adaptive backstepping synchronization of uncertain chaotic system.
Chaos, Solitons and Fractals 21 (2004) 643–649.
Sinha SC, Henrichs JT, Ravindra BA. A general approach in the design of active controllers for
nonlinear systems exhibiting chaos. Int J Bifurcat Chaos 2000;10(1):165–78.
M.T. Yassen. Chaos control of chaotic dynamical systems using backstepping design. Chaos, Solitons
and Fractals 27 (2006) 537–548.
Ali Reza Sahab and Mohammad Haddad Zarif. Improve Backstepping Method to GBM. World
Applied Sciences Journal 6 (10): 1399-1403, 2009, ISSN 1818-4952.
Sahab, A.R. and M. Haddad Zarif. Chaos Control in Nonlinear Systems Using the Generalized
Backstopping Method. American J. of Engineering and Applied Sciences 1 (4): 378-383, 2008, ISSN
1941-7020.
Ali Reza Sahab, Masoud Taleb Ziabari, Seyed Amin Sadjadi Alamdari. Chaos Control via Optimal
Generalized Backstepping Method. International Review of Electrical Engineering (I.R.E.E), Vol.5,
n.5.
Sundarapandian Vaidyanathan, CONTROLLING HYPERCHAOS AND SYNCHRONIZATION OF
AN UNCERTAIN MODIFIED HYPERCHAOTIC LÜ SYSTEM, International Journal of
Instrumentation and Control Systems (IJICS) Vol.2, No.1, January 2012.
Sundarapandian Vaidyanathan, OUTPUT REGULATION OF SPROTT-G CHAOTIC SYSTEM BY
STATE FEEDBACK CONTROL, International Journal of Instrumentation and Control Systems
(IJICS) Vol.1, No.1, July 2011.
Sundarapandian Vaidyanathan, ADAPTIVE CONTROL AND SYNCHRONIZATION OF
HYPERCHAOTIC CAI SYSTEM, International Journal of Control Theory and Computer Modelling
(IJCTCM) Vol.1, No.1, July 2011.
Sundarapandian Vaidyanathan, OUTPUT REGULATION OF THE SIMPLIFIED LORENZ
CHAOTIC SYSTEM, International Journal of Control Theory and Computer Modelling (IJCTCM)
Vol.1, No.3, November 2011.
Sundarapandian Vaidyanathan, STATE FEEDBACK CONTROLLER DESIGN FOR THE OUTPUT
REGULATION OF SPROTT-H SYSTEM, International Journal of Information Sciences and
Techniques (IJIST) Vol.1, No.3, November 2011.
Sundarapandian Vaidyanathan, ACTIVE CONTROLLER DESIGN FOR THE OUTPUT
REGULATION OF SPROTT-K CHAOTIC SYSTEM, Computer Science & Engineering: An
International Journal (CSEIJ), Vol.2, No.3, June 2012.
Dr. V. Sundarapandian, Adaptive Control and Synchronization of the Uncertain Sprott H System,
International Journal of Advances in Science and Technology, Vol. 2, No.4, 2011.
Sprott, J.C, Some simple chaotic flows, Phys. Rev. E., Vol. 50, pp 647-650.
87](https://image.slidesharecdn.com/adaptivetrackingcontrolofsprott-hsystem-131207003851-phpapp02/85/Adaptive-tracking-control-of-sprott-h-system-5-320.jpg){kind=tracking}
Adaptive tracking control of sprott h system
- 1. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013 ADAPTIVE TRACKING CONTROL OF SPROTT-H SYSTEM Masoud Taleb Ziabari 1 and Ali Reza Sahab 2 1 2 Faculty of Engineering, Computer Engineering Group, Ahrar University, Rasht, Iran. Faculty of Engineering, Electrical Engineering Group, Islamic Azad University, Lahijan Branch, Iran. Abstract This paper investigates the tracking problem of Sprott-H chaotic system which is three-dimensional chaotic systems discovered by Sprott (1994). New nonlinear control laws are derived for the tracking problem of uncertain Sprott-H chaotic system with unknown parameters. The adaptive generalized backstepping method is applied to control of uncertain Sprott-H chaotic system. Numerical simulations are presented to demonstrate the effectiveness of the control schemes. KEYWORDS Sprott-H Chaotic System, Uncertain, Adaptive Generalized Backstepping Method, Control. 1. INTRODUCTION In recent years, various controllers have been proposed to achieve the control of chaotic systems [3-9]. The adaptive synchronization of an uncertain modified hyperchaotic Lü system was investigated in [10]. In [11], the output regulation problem for the Sprott-G chaotic system (1994) has been studied in detail. The stabilization and synchronization of the hyperchaotic Cai system with unknown system parameters was applied by adaptive control theory. The feedback controllers for control of the simplified Lorenz system was investigated in [13]. [14] derive state feedback controllers for the output regulation problem of the Sprott-H chaotic system (1994). In [15], active controller has been designed to solve the output regulation problem for the Sprott-K chaotic system (1994) and a complete solution for the tracking of constant reference signals (setpoint signals). [16] applied adaptive control theory for the control and synchronization of the Sprott H chaotic system (Sprott, 1994) with unknown system parameters. The rest of the paper is organized as follows: In section 2, Sprott-H system is presented. In section 3, the generalized backstepping method is studied. In section 4, the generalized backstepping controller is designed for racking any desired inputs. In section 5, Represents simulation results. Finally, in section 6, Provides conclusion of this work. DOI : 10.5121/ijitmc.2013.1409 83
- 2. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013 2. SYSTEM DESCRIPTION The three-dimensional Sprott-H chaotic system [17] described by the dynamics ̇ ̇ (1) ̇ Where are positive constants and are variables of the system, when system (1) is chaotic. See Figure 1 and Figure 2. , the 6 x y z Trajectory of States 4 2 0 -2 -4 -6 0 100 200 300 400 500 600 Time (sec) 700 800 900 1000 Figure 1. Time response of the system (1). 2 1 z 0 -1 -2 -3 6 4 2 0 2 -2 0 y -4 -2 -6 x Figure 2. Phase portraits of the hyperchaotic attractors (1). 3. GENERALIZED BACKSTEPPING METHOD Generalized backstepping method [7-9] is applied to nonlinear systems as follow { ̇ ̇ Where ( ) ( ) ( ) ( and [ (2) ) ] . Suppose the function ( ) is the lyapunov function. 84
- 3. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013 ( ) ∑ (3) The control signal and the extended lyapunov function of system (2) are obtained by equations (4),(5). ( ( ∑ { ) ) ∑ [ ( ) ∑ ( ) ∑ ∑ [ ∑ [ ( ) ] ( )] ( } ) ( )] (4) (5) 4. TRACKING OF CHAOTIC SYSTEM First, we will add control laws and the desired trajectory ( ). ̅ ( ̅ ̇ to the equation (1) let ̅ be the deviation between the output ( ) ) ̇ (6) ̇̅ ̅ ̇ Stabilization of the state: In order to use the theorem, it is sufficient to establish equation (7). ( ̅) ̇ ( ̅) (7) According to the theorem, the control signals will be obtained from the equations (8). ( ) ( ) ̂ ̂ ( ) (8) The parameters are unknown and ̂ ̂ are respectively estimated values of parameters which are updated by following equation. ̂̇ ̂̇ ̂ (9) ̂ And Lyapunov function as ( ) ( ̅ ) ( ) (10) we select the gains of controllers (8) in the following form (11) 5. NUMERICAL SIMULATION This section presents numerical simulations Sprott-H chaotic system. The generalized backstepping method (GBM) is used as an approach to control chaos in Sprott-H chaotic system. ( ) ( ) The initial values of the Sprott-H chaotic system are ( ) . Case 1 : Tracking ( ) , 85
- 4. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013 Case 2 : Tracking ( ) Case 3 : Tracking ( ) , . Figure 3 shows that the scalar output can track the Step Input with the control inputs (8). Figure 4 ( ). Figure 5 shows that shows that the scalar output can track the desired trajectory ( ) the scalar output can track the desired trajectory ( ) . 1 Trajectory of Output 0.5 0 -0.5 -1 -1.5 -2 0 1 2 3 4 5 6 Time (sec) 7 8 9 10 Figure 3. The time response of signal ( ) for tracks the Step Input. 1 Trajectory of Output 0.5 0 -0.5 -1 -1.5 -2 0 5 10 15 Time (sec) 20 25 30 Figure 4. The time response of signal ( ) for tracks the desired trajectory ( ) ( ). 2 1.5 Trajectory of Output 1 0.5 0 -0.5 -1 -1.5 -2 0 1 2 3 4 5 6 Time (sec) 7 8 9 10 Figure 5. The time response of signal ( ) for tracks the desired trajectory ( ) . 86
- 5. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013 6. CONCLUSIONS In this paper, we applied adaptive generalized backstepping control for the control of the Sprott-H chaotic system with unknown system parameters. The tracking problem considered for the SprottH chaotic system was for the tracking of any desired input signals. Numerical simulations show that the proposed method work effectively. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] Chao-Chung Peng, Chieh-Li Chen. Robust chaotic control of Lorenz system by backstepping design. Chaos, Solitons and Fractals 37 (2008) 598–608. Cheng-Chi Wang, Neng-Sheng Pai, Her-Terng Yau. Chaos control in AFM system using sliding mode control by backstepping design. Commun Nonlinear Sci Numer Simulat 15 (2010) 741–751. Faqiang Wang, Chongxin Liu. A new criterion for chaos and hyperchaos synchronization using linear feedback control. Physics Letters A 360 (2006) 274–278. Yongguang Yu, Suochun Zhang. Adaptive backstepping synchronization of uncertain chaotic system. Chaos, Solitons and Fractals 21 (2004) 643–649. Sinha SC, Henrichs JT, Ravindra BA. A general approach in the design of active controllers for nonlinear systems exhibiting chaos. Int J Bifurcat Chaos 2000;10(1):165–78. M.T. Yassen. Chaos control of chaotic dynamical systems using backstepping design. Chaos, Solitons and Fractals 27 (2006) 537–548. Ali Reza Sahab and Mohammad Haddad Zarif. Improve Backstepping Method to GBM. World Applied Sciences Journal 6 (10): 1399-1403, 2009, ISSN 1818-4952. Sahab, A.R. and M. Haddad Zarif. Chaos Control in Nonlinear Systems Using the Generalized Backstopping Method. American J. of Engineering and Applied Sciences 1 (4): 378-383, 2008, ISSN 1941-7020. Ali Reza Sahab, Masoud Taleb Ziabari, Seyed Amin Sadjadi Alamdari. Chaos Control via Optimal Generalized Backstepping Method. International Review of Electrical Engineering (I.R.E.E), Vol.5, n.5. Sundarapandian Vaidyanathan, CONTROLLING HYPERCHAOS AND SYNCHRONIZATION OF AN UNCERTAIN MODIFIED HYPERCHAOTIC LÜ SYSTEM, International Journal of Instrumentation and Control Systems (IJICS) Vol.2, No.1, January 2012. Sundarapandian Vaidyanathan, OUTPUT REGULATION OF SPROTT-G CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL, International Journal of Instrumentation and Control Systems (IJICS) Vol.1, No.1, July 2011. Sundarapandian Vaidyanathan, ADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC CAI SYSTEM, International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.1, July 2011. Sundarapandian Vaidyanathan, OUTPUT REGULATION OF THE SIMPLIFIED LORENZ CHAOTIC SYSTEM, International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.3, November 2011. Sundarapandian Vaidyanathan, STATE FEEDBACK CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF SPROTT-H SYSTEM, International Journal of Information Sciences and Techniques (IJIST) Vol.1, No.3, November 2011. Sundarapandian Vaidyanathan, ACTIVE CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF SPROTT-K CHAOTIC SYSTEM, Computer Science & Engineering: An International Journal (CSEIJ), Vol.2, No.3, June 2012. Dr. V. Sundarapandian, Adaptive Control and Synchronization of the Uncertain Sprott H System, International Journal of Advances in Science and Technology, Vol. 2, No.4, 2011. Sprott, J.C, Some simple chaotic flows, Phys. Rev. E., Vol. 50, pp 647-650. 87